How does "gaining the right" to advance classes work?

As I mentioned in this question, class changes are done at Shrines of Mila, rather than via Seals, so I'm at a loss for the way class changes work in this game.

That said, I've noticed some disparity in the way units have been "gaining the right" to advance classes. For instance, Lukas gained the right to the Knight class once he hit level 7 as a Soldier, similar to the way Tobin gained the right to the Myrmidon class at level 7 as a Mercenary. However, Alm has yet to gain the right to advance past the Fighter class (it is the the same "level" as the Soldier and Mercenary classes), despite being level 8. Faye also has yet to gain the right to a class higher than Pegasus Knight (she's level 8, as well).

Is there some sort of criteria for gaining the right to different classes? If I choose not to change classes right away, will I eventually get more options?